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Hierarchical Microstructures Self-Assembled from Linear Multiblock Copolymers in Thin Films Jingyuan Lin, Shuting Gong, Xu Zhang, Liquan Wang* Hierarchical lamellae self-assembled from linear multiblock copolymers in thin films are investigated by self-consistent field theory. The thin films are confined between two parallel substrates. The confinement strategy allows generating hierarchical microstructures with various numbers and different orientations of small-length-scale lamellae. Effects of film thickness and surface affinity on the structures are studied. It is found that not only the period of the large- and small-length-scale lamellae but also the orientation of small-length-scale lamellae relative to large-length-scale lamellae can be tuned by varying the film thickness. Moreover, the structures of the hierarchical lamellae can be tailored by changing the surface affinity. Through analyzing free energies of various lamellae, phase diagrams are mapped out. The present work could provide guidance for fabricating hierarchical microstructures in a controllable way.
1. Introduction Advanced development of nanotechnologies, in fields such as integrated circuit, requires fabrication of materials possessing multiple nanometer length scales.[1] Recent efforts have been greatly devoted to create hierarchical microstructures, which is at multiple length scales, by virtue of the self-assembly of precise building molecules.[2–10] One of the building molecules concerns a linear multiblock copolymer consisting of one or two tails and many midblocks.[5–10] The number of midblocks, the length of tail blocks, and the degree of repulsion control the selfassembled hierarchical microstructures, which have double
J. Lin School of Marine Science, Shanghai Ocean University, Shanghai 201306, China S. Gong, X. Zhang, L. Wang Shanghai Key Laboratory of Advanced Polymeric Materials, State Key Laboratory of Bioreactor Engineering, Key Laboratory for Ultrafine Materials of Ministry of Education, School of Materials Science and Engineering, East China University of Science and Technology, Shanghai 200237, China E-mail:
[email protected]
468
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periodicities including the large period of multiblock copolymers and the small period of multiblocks. Despite of the advancement of creating microstructures at multiple length scales, it is far from meeting the demands in practical applications because diversified molecules should be designed to produce the needed microstructures. Generally, one would like to obtain the required multiple-length-scale microstructures without having to change the chemistry of synthetical multiblock copolymers. Manipulation of fabricating method could be helpful to generate various hierarchical microstructures with specific multiblock copolymer chemistry. A promising technique relies on two parallel substrates to drive the organization of linear multiblock copolymers to a variety of frustrated microstructures in thin films. This technique has been widely applied to direct the self-assembly of block copolymers.[11–16] The orientation and periodicities of microstructures can be tailed by the distance and the affinity of two substrates. By introducing the substrates into linear multiblock copolymer systems, expect for the modulation of large-length-scale structures, the number as well as orientation of small-length-scale structures is expected to be tunable. In our previous work, we performed dissipative particle dynamics (DPD) simulations on the self-
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DOI: 10.1002/mats.201500031
Hierarchical Microstructures Self-Assembled. . . www.mts-journal.de
assembly of linear multiblock copolymers in thin films.[17] The work demonstrates that not only the number of large- or small-length-scale lamellae but also the orientation of the small-length-scale lamellae relative to large-length-scale lamellae can be tuned by the film thickness. However, due to the limitation of DPD simulations, the stability of the hierarchical lamellae at various film thicknesses remains less well-understood. In addition, other effects such as surface affinity on the selfassembly behavior haven’t been investigated. In this work, we took advantage of self-consistent field theory (SCFT) to address these problems. The SCFT, a widely-used theory for predicting equilibrium phases of complex fluids, can provide an insight into the stability of the microstructures of copolymers.[18–27] For example, the SCFT has been extended to investigate the selfassembly of linear multiblock copolymers.[28–32] The works not only reproduce the experimental findings of hierarchical microstructures, but also predict a series of new microstructures. In addition to the bulk, the SCFT was also applied for the self-assembly of block copolymers in thin films. Recently, Fredrickson et al. developed a ‘‘masking’’ technique that confines the block copolymers between walls by choosing a ‘‘masking’’ function to fit the geometry
with diversified hierarchical microstructures in a controllable way.
2. Theoretical Method We consider an incompressible melt of monodisperse A(BC)n linear multiblock copolymers confined between two parallel solid substrates, separated by a distance D, as shown in Figure 1. The A(BC)n linear multiblock copolymer is comprised of an end A-block connected to a (BC)nmultiblock consisting of n elementary diblock repeat units. The total volume fractions of A-, B-, and C-blocks are fA, fB, and fC, respectively. The lengths of B- and C-blocks are assumed to be equal and the volume fractions are specified 1f by Df B ¼ Df C ¼ n A . The substrates are assumed to consist of nW wall ‘‘particles’’ in a volume V. Within the SCFT framework, one considers the conformations of a single copolymer chain in a set of effective chemical potential field vI ðrÞ, where I denotes A-, B-, and C-species. We invoke an incompressibility (fA ðrÞ þ fB ðrÞ þ fC ðrÞ þ fW ðrÞ ¼ 1) by introducing a Lagrange multiplier jðrÞ. The free energy per chain is given by
8 9 > > > !> > > Z > nkB T V V 2 > > I¼A;B;C I¼A;B;C;W > > : I; J ¼ A; B; C; W ;
ð1Þ
I 6¼ J
of the walls.[33] This technique is particularly suited for the numerical implement of the SCFT equations by pseudospectral method. Due to its numerical efficiency, this SCFT has been widely used to study the self-assembly of block copolymers confined in the space with various geometries.[34–37] In the present work, we used the SCFT to undertake an investigation on the hierarchical microstructures self-assembled from A(BC)n linear multiblock copolymers in thin films between two solid substrates. The parallel substrates were built upon the ‘‘masking’’ technique proposed by Fredrickson et al. The influences of the film thickness and the surface affinity on the hierarchical microstructures were examined. The stability of the microstructures was analyzed by comparing the free energies. The hierarchical microstructures were found to exhibit not only a variation of the number of the large- and small-length-scale lamellae but also a transition from parallel to perpendicular lamellae-inlamellae, as the film thickness increases. The results could provide the guidance for fabricating thin films
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where xIJ is the Flory-Huggins parameter between I- and Jspecies. Q is the partition function of a single chain in the effective chemical potential field vI ðrÞ (I ¼ A, B, and C) in terms of propagators qðr; sÞ and qþ ðr; sÞ. The spatial coordinate r is rescaled by Rg, where R2g ¼ a2 N=6. The
Figure 1. Sketch of A(BC)3 linear multiblock copolymer thin films confined between two solid substrates. The D represents film thickness. The blue, red, and green colors are assigned to A-, B-, and C-blocks, respectively.
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contour length is parameterized with variable s, which starts from one end (s ¼ 0) to the other (s ¼ 1). The propagators qðr; sÞ and qþ ðr; sÞ satisfy following modified diffusion equations i @qðr; sÞ h 2 2 ¼ Rg r vuðsÞ ðrÞ qðr; sÞ @s i @qþ ðr; sÞ h 2 2 ¼ Rg r vuðsÞ ðrÞ qþ ðr; sÞ @s
ð2Þ
with the initial condition qðr; 0Þ ¼ 1 and qþ ðr; 1Þ ¼ 1, respectively. Here, uðsÞ is used to specify the segment type along the copolymer chain, subject to (0 i < n) 8 >
: C
if if if
Z
fA
V Q
fB ðrÞ ¼
n1 VX Q i¼0
fC ðrÞ ¼
n1 VX Q i¼0
dsqðr; sÞqþ ðr; sÞ
ð4Þ
0
Z
f A þiðDf B þDf C ÞþDf B f A þiðDf B þDf C Þ
Z
f A þðiþ1ÞðDf B þDf C Þ f A þiðDf B þDf C ÞþDf B
dsqðr; sÞqþ ðr; sÞ
dsqðr; sÞqþ ðr; sÞ
ð5Þ
ð6Þ
Minimization of free energy F, with respect to fA ðrÞ, fB ðrÞ, fC ðrÞ, and jðrÞ, can lead to a set of mean-field equations, vA ðrÞ ¼ xAB NfB ðrÞ þ xAC NfC ðrÞþxAW NfW ðrÞ þ jðrÞ
ð7Þ
vB ðrÞ ¼ xAB NfA ðrÞ þ xBC NfC ðrÞ þ xBW NfW ðrÞ þ jðrÞ
ð8Þ
vC ðrÞ ¼ xAC NfA ðrÞ þ xBC NfB ðrÞ þ xCW NfW ðrÞ þ jðrÞ
ð9Þ
fA ðrÞ þ fB ðrÞ þ fC ðrÞ þ fW ðrÞ ¼ 1
ð10Þ
The density field of wall ‘‘particles’’ fW ðrÞ is a fixed function of r that is specified before starting the SCFT simulations, following the ‘‘masking‘‘ technique proposed
470
1 dðrÞ 1 þ tanh a fW ðrÞ ¼ 2 e
ð11Þ
Here, a and e are factors used to define the transition region and set the width of the transition region, respectively; d(r) is the distance from the point r to the nearest edge of the boundary of the wall. We assume that the boundary of the wall is at fW ðrÞ ¼ 0:5, and select a ¼ logð99Þ ¼ 4:5951 such that the wall transition region
0 < s < fA f A þ i Df B þ Df C < s < f A þ i Df B þ Df C þ Df B f A þ i Df B þ Df C þ Df B < s < f A þ ði þ 1Þ Df B þ Df C
The partition function, Q, for an unconstrained chain is R evaluated by Q ¼ drqðr;1Þ. The segment densities fA ðrÞ, fB ðrÞ, and fC ðrÞ follow that fA ðrÞ ¼
by Fredrickson et al.[33] and extended by others.[34–36] The transition of fW ðrÞ is selected to be a hyperbolic tangent form, which is as follows
ð3Þ
begins at fW ðrÞ ¼ 0:01 and ends at fW ðrÞ ¼ 0:99. The e is set to be 0.5Rg which is approximately equal to the interface width. As suggested by Fredrickson et al., this e value couldn’t affect the results.[33] This ‘‘masking’’ technique can retain the stability characteristics of standard saddle point search methods and is particularly suited for the numerical implement of the SCFT equations by pseudospectral method. To solve the SCFT equations, we use a variant of the algorithm developed by Fredrickson and co-workers.[18,19] The calculations were started from a random initial state. When analyzing free energies of various lamellae, the calculations were started from a deterministic initial field constructed from the structures observed form the simulations initiated form a random field. The diffusion equations were solved with the fourth-order backward differentiation formula, which has higher accuracy and stability for strongly segregated systems.[38,39] The densities fI ðrÞ of I-species, conjugated the potential fields vI ðrÞ, are evaluated by Eqs. (4)–(10). The potential fields vI ðrÞ are updated by using a two-step Anderson mixing scheme.[40] The simulations in this work can be carried out in the space with periodic boundary conditions owing to the ‘‘masking‘‘ technique. In the calculations, the spatial resolutions were taken as 0:01Rg < Dr < 0:07Rg . The contour step sizes in the simulation were set at Ds ¼ 0:001, which is sufficient to guarantee the accuracy. The numerical simulations proceeded until the relative free energy changes at each iteration were smaller than 108 and the incompressibility condition was achieved. To obtain the stable perpendicular lamellae-in-lamellae at a fixed film thickness, we optimized the free energy with respect to the size of unconfined directions, as suggested by Bohbot-Raviv and Wang.[41] While for the parallel lamellae-
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in-lamellae at a fixed film thickness, the free energy does not change as a function of the size of unconfined directions, and therefore it is unnecessary to minimize the free energy with respect to the size.
3. Results and Discussion In this work, we mainly focused on the effect of film thickness and surface affinity on the hierarchical lamellae formed by A(BC)3 linear multiblock copolymers. To ensure the formation of hierarchical lamellae, the length of Ablocks was set to equal that of (BC)3-multiblocks, and thus fA ¼ 0.50 and fB ¼ fC ¼ 0.25. In our previous studies for the self-assembly of A(BC)3 linear multiblock copolymers in bulk, we found that the formed hierarchical lamellae can be either perpendicular or parallel lamellae-in-lamellae, depending on the interaction strength.[28] In this work, we carried out further calculations for the bulk phases of A(BC)3 linear multiblock copolymers. We found that the perpendicular lamellae-in-lamellae in which the smalllength-scale lamellae is normal to the large-length-scale lamellae are formed at xABN ¼ 200, xACN ¼ 100, and xBCN ¼ 400, and the parallel lamellae-in-lamellae where the large- and small-length-scale lamellae are parallel each other are stable at xABN ¼ 300, xACN ¼ 200, and xBCN ¼ 200. Note that five BCBCB-layers were included in the parallel lamellae-in-lamellae. The periods D of perpendicular lamellae-in-lamellae and parallel lamellae-in-lamellae are 5.95Rg and 6.5Rg, respectively. The first corresponds to frustrated cases, while the second corresponds to nonfrustrated cases.[28] In the frustrated cases, the xABN is greater than the xACN, and therefore the A/C-contact becomes more energetically favored than the A/B-contact in terms of interaction enthalpy. It is possible for the A/C-interface to be formed, although there is no chain junction between A- and C-blocks. In the nonfrustrated cases, the xABN is less than or equal to the xACN. Under this circumstance, the interactions between the A- and B-blocks are less thermodynamically costly than those of A- and C-blocks, which is unfavorable for the A/C-interface to be produced. In what follows, we carried out studies on the thin films of these two cases, respectively. 3.1. Thin Films from Self-Assembly of Frustrated Linear Multiblock Copolymers We first studied the frustrated systems of A(BC)3 linear multiblock copolymers confined in thin films. The interaction strengths are set as xABN ¼ 200, xACN ¼ 100, and xBCN ¼ 400. In the bulk, the linear multiblock copolymers self-assemble into the perpendicular lamellae-in-lamellae. In contrast to the bulk, the structures of A(BC)3 linear multiblock copolymers in thin films can be much richer.
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Figure 2. Single-period hierarchical lamellae of (a) L31, (b) L?1, (c) L51, and (d) L71 self-assembled from A(BC)3 linear multiblock copolymers with fA ¼ 0.50, fB ¼ fC ¼ 0.25, xABN ¼ 200, xACN ¼ 100, xBCN ¼ 400 in thin films. The solid substrates are selective to A-blocks by setting xAWN ¼ 0 and xBWN ¼ xCWN ¼ 150. The colors appear as the same as those in Figure 1.
Figure 2 shows the lamellae-in-lamellae formed in the films with small film thickness D. Since the films are very thin, the lamellae-in-lamellae with single period of large-lengthscale lamellae could be observed. These singly periodic hierarchical lamellar structures include parallel lamellaein-lamellae with three BCB-layers (L31, Figure 2a), perpendicular lamellae-in-lamella (L?1, Figure 2b), parallel lamellae-in-lamella with five BCBCB-layers (L51, Figure 2c), and parallel lamellae-in-lamella with seven BCBCBCB-layers (L71, Figure 2d). Note that in the representations such as L31 and L?1, the first bold letter L, subscripts, and last number denotes the lamellar structures, the number of the parallel packed small-length-scale lamellae (symbol ? means that the small-length-scale lamellae are normal to the largelength-scale lamellae), and the number of the large-lengthscale structures, respectively. The definition of other representations also follows this rule. With increasing the film thickness, the period of largelength-scale lamellae increases. Figure 3 shows the representative lamellae-in-lamellae assembled in the relatively thicker films. As can be seen, the large-lengthscale period increases from single to double, while the small-length-scale structures appear as the same as those of single periodic hierarchical lamellae (see Figure 2). As a result, the observed double-period hierarchical lamellae could be doubly periodic parallel lamellae-in-lamellae with three BCB-layers (L32, Figure 3a), doubly periodic perpendicular lamellae-in-lamella (L?2, Figure 3b), doubly periodic parallel lamellae-in-lamella with five BCBCB-layers (L52, Figure 3c), and doubly periodic parallel lamellae-in-lamella with seven BCBCBCB-layers (L72, Figure 3d). To gain insight into the stability of these lamellae, we examined the free energies of the lamellae at various film thicknesses. The free energies plotted as a function of film thickness are presented in Figure 4. As shown in Figure 4, the free energies of parallel lamellae-in-lamellae with three BCB-layers, including L31 and L32, are much higher than
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Figure 3. Double-period hierarchical lamellae of (a) L32, (b) L?2, (c) L52, and (d) L72 self-assembled from A(BC)3 linear multiblock copolymers with fA ¼ 0.50, fB ¼ fC ¼ 0.25, xABN ¼ 200, xACN ¼ 100, xBCN ¼ 400 in thin films. The solid substrates are selective to A-blocks by setting xAWN ¼ 0 and xBWN ¼ xCWN ¼ 150. The colors appear as the same as those in Figure 1.
those of other structures (see blue lines). This indicates that the parallel lamellae-in-lamellae with three BCB-layers are unstable. For very thin films, the L31 is much more unstable than L?1 (red line). This is due to the fact that the B- and Cblocks in perpendicular lamellae-in-lamellae suffer from less marked confinement imposed by the substrates because the small-length-scale lamellae can orient perpendicularly to the confined layers to retain bulk period. In addition, the L71 is more stable than L32, since the B- and C-blocks can be more relaxed in the L71. Compared with the free energies of L?1 and L51, we noted that the free energy of L?1 is smaller than that of L51 as the film thickness is smaller than about 6.5Rg or larger than 8.7Rg. Similar situation also exists for the L?2 and L52, where the L?2 can be more stable than the L52 as the film thickness is smaller than about 12.6Rg or larger than 16.4Rg. By comparing the free energies, we mapped out the stability regions of the lamellae-in-lamellae. The diagram is
35.0
N=200
AB
L31
N=400
16.0 14.0
L52
L1 L2
L51
6.0
L72
/Rg
F/nkBT
AC
L71
32.0
12.0 10.0 8.0
9.0
12.0
15.0
18.0
/Rg
6.0 4.0
Figure 4. Free energy F/nkBT as a function of the film thickness D/Rg for various lamellae-in-lamellae. The parameters for the copolymers are fA ¼ 0.50, fB ¼ fC ¼ 0.25, xABN ¼ 200, xACN ¼ 100, xBCN ¼ 400. The solid substrates are selective to A-blocks by setting xAWN ¼ 0 and xBWN ¼ xCWN ¼ 150.
472
18.0
N=100
L32
BC
34.0 33.0
given in Figure 5. As can be seen from Figure 5, a phase transition of L?1 ! L51 ! L?1 ! L71 ! L?2 ! L52! L?2 ! L72 occurs as the film thickness increases. Note that not only the period of large- and small-length-scale lamellae increases, but also a phase transition from parallel to perpendicular hierarchical lamellae takes place, as the film thickness increases. Interestingly, reentrant phase transitions of the perpendicular lamellae-in-lamellae were observed. For example, the L?1 can be not only formed at D ¼ 4.16.5Rg, but also formed at D ¼ 8.79.0Rg. An intuition is that the L?1 can only be formed at smaller D. This is based on the consideration that the B- and C-blocks can be relaxed in these structures due to the smaller space constrained between two substrates. The formation of the L?1 at larger D counters with the intuition. However, the regions of perpendicular lamellae-inlamellae at larger film thickness are much narrower than those at smaller film thickness. Due to the narrower regions for the reentrant perpendicular lamellae-inlamellae, such phenomenon was not revealed in our previous work by DPD simulations.[17] To further understand the reentrant phase transitions, we calculated the enthalpic and entropic contributions to the free energy for the L?1 and L51 structures.[28,42] The results are presented in Figure 6. As can be seen from Figure 6a, in the transition from L?1 to L51 at about D ¼ 6.5Rg, the entropic losses for both structures show a slight increase. However, the enthalpy for L51 shows a shaper decrease than that for L?1 as the D increases (see Figure 6b). This implies that the formation of L51 as D > 6.5Rg can reduce unfavorable enthalpy effectively. Moreover, the enthalpy for L?1 is smaller than that for L51, suggesting that the stability of L?1 as D < 6.5Rg can be ascribed to its lower enthalpy. In the transition from L51 to L?1 at about D ¼ 8.7Rg, the enthalpies for both the structures show a slight decrease, but the entropic loss for L?1 increases slower than that for L51. As a result, the L51 ! L?1
A B C
L72 L52
N=200
AB
L2
N=100
AC
N=400
L71
BC
L51
AW
N=0 N=150
BW
L1
N=150
CW
Figure 5. Stability regions of lamellae-in-lamellae for A(BC)3 linear multiblock copolymer thin films. The parameters for the copolymers are fA ¼ 0.50, fB ¼ fC ¼ 0.25, xABN ¼ 200, xACN ¼ 100, xBCN ¼ 400. The solid substrates are selective to A-blocks by setting xAWN ¼ 0 and xBWN ¼ xCWN ¼ 150.
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23.5
(a)
L1
N= N
23.0
L1 L51
100
L1
N=
CW
22.5 22.0
L71
BW
-S/nKB
150
L1
L51
L51
L2
L1
21.5 6.0 6.2 6.4 6.6 6.8 8.4 8.6 8.8 9.0 /Rg
12.0
(b)
L1
L51
L1
U/nKBT
11.5 11.0 10.5 L51
10.0
L1
6.0 6.2 6.4 6.6 6.8 8.4 8.6 8.8 9.0 /Rg Figure 6. (a) Entropic loss –S/nKB and (b) enthalpy U/nkBT and as a function of the film thickness D/Rg for L51 and L?1. The parameters are fA ¼ 0.50, fB ¼ fC ¼ 0.25, xABN ¼ 200, xACN ¼ 100, xBCN ¼ 400. The solid substrates are selective to A-blocks by setting xAWN ¼ 0 and xBWN ¼ xCWN ¼ 150. The green dash lines indicate where the phase transitions occur.
transition is dominated by the entropy. In addition, the entropic loss for L?1 is smaller than that for L51, and thus the appearance of L?1 as D > 8.7Rg is entropic favored. In addition, we carried out a further study on the effect of surface affinity on the lamellae-in-lamellae formed by the linear multiblock copolymers. The data were summarized into a phase diagram in space of xN versus D. Here, the interaction strength xN denotes xBWN that equals xCWN. In the study, the xN is varied from 50 to 150. Since the xAWN is fixed as 0, the substrates are always selective to the Ablocks. The diagram is shown in Figure 7. As can been seen, the xN shows a less marked influence on the phase transitions, except for the phase transition of L?1 ! L51 ! L?1. With increasing the xN value, the regions of L?1 first broaden and then narrow. This result indicates that the structures of lamellae-in-lamellae are less dependent on the surface affinity as long as the xN is high enough.
3.2. Thin Films from Self-Assembly of Nonfrustrated Linear Multiblock Copolymers In addition to the thin films of frustrated linear multiblock copolymers, the thin films formed by nonfrustrated A(BC)3
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L52
50 6.0
8.0
10.0
12.0
14.0
/Rg Figure 7. Phase diagram in space of xN versus D/Rg for A(BC)3 linear multiblock copolymer thin films. The parameters for the copolymers are fA ¼ 0.50, fB ¼ fC ¼ 0.25, xABN ¼ 200, xACN ¼ 100, xBCN ¼ 400. The solid substrates are selective to A-blocks by setting xAWN ¼ 0.
linear multiblock copolymers were also studied. The interaction strengths are set as xABN ¼ 300, xACN ¼ 200, and xBCN ¼ 200. In this case, the parallel lamellae-inlamellae are formed in the bulk. We first studied the selfassembly of nonfrustrated linear multiblock copolymers confined between two solid substrates that are selective to A-blocks by setting xAWN ¼ 0 and xBWN ¼ xCWN ¼ 100. The results are shown in Figure 8. Figure 8a shows the free energies of various hierarchical lamellae as a function of film thickness. As shown in Figure 8a, the free energies of the lamellae-in-lamellae with three BCB-layers such as L31 and L32 are higher than those of other lamellae, indicating that these hierarchical lamellae are unstable, similar to the thin films of frustrated case (see blue lines in Figure 4). The L?2 structures (red line), whose free energy is higher than that of L52 and other lamellae, were also found to be unstable. Moreover, the free energy of L?1 is smaller than that of L51 as the D is smaller than 5.8Rg and larger 7.5Rg. Based on the free energy analysis of various lamellar structures, the stability regions of the lamellae-in-lamellae were mapped out, which is shown in Figure 8b. As the film becomes thick, a phase transition of L?1 ! L51 ! L?1 ! L71 ! L52 ! L72 takes place. The reentrant phase transition of L?1 was also found. Moreover, the region of L?1 at larger film thickness enlarges, compared with the thin films of frustrated case. Note that the L?2 disappears in the phase transition, probably due to following reason. As the film becomes thick, the self-assembly of linear multiblock copolymers in the film could be similar to that in the bulk. In the bulk, the nonfrustrated linear multiblock copolymers don’t favor the formation of perpendicular lamellae-inlamellae. Provided that the L?2 is formed as the film becomes thick, the enthalpy could be much more unfavorable than the other lamellae owing to unfavorable interaction between A- and C-blocks from the viewpoint of free energies. As a result, the L?2 disappears in this case.
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31.0
(a)
16.0
N=200
14.0
BC
/Rg
F/nkBT
N=200
AC
L31
29.0 L2
28.0 27.0
18.0 16.0 14.0 /Rg
AB
L51
30.0
12.0
L1
6.0
(b)
L71
9.0
12.0 /Rg
15.0
A B C N=200 N=200
BC
8.0
L1 L51
AW
N=0 N=100
BW
N=100
CW
Figure 8. (a) Free energy F/nkBT as a function of the film thickness D/Rg for various lamellae-in-lamellae. (b) Stability regions of lamellae-in-lamellae for A(BC)3 linear multiblock copolymer thin films. The parameters for the copolymers are fA ¼ 0.50, fB ¼ fC ¼ 0.25, xABN ¼ 300, xACN ¼ 200, xBCN ¼ 200. The solid substrates are selective to A-blocks by setting xAWN ¼ 0 and xBWN ¼ xCWN ¼ 100.
We further investigated the self-assembly behavior of linear multiblock copolymers confined between two substrates that are nonselective to A-, B-, and C-blocks. Without the loss of generality, the interaction strengths between the substrates and multiblock copolymers were set as xAWN ¼ xBWN ¼ xCWN ¼ 0. Figure 9 shows the stability regions of lamellae-in-lamellae at various film thicknesses. As can be seen, a series of new lamellae-inlamellae that are reversal of the lamellae-in-lamellae shown in Figure 2 and 3 were observed. In these microstructures, the B- and C-blocks are absorbed to the substrates due to the entropic reason.[15] Since the lamellae-in-lamellae lose the periodicity along the directions normal to the substrates, the hierarchical lamellae shown in Figure 9 are completely different from those shown in Figure 2 and 3. The stable lamellar microstructures include singly periodic perpendicular lamellaein-lamellae (R?1), singly periodic parallel lamellae-inlamellae with three BCB-layers near substrates (R31), singly periodic parallel lamellae-in-lamellae with four BCBClayers near substrates (R41), doubly periodic parallel lamellae-in-lamellae with three BCB-layers near substrates and five BCBCB-layers in the middle (R352), doubly periodic parallel lamellae-in-lamellae with three BCB-layers near
474
R372 R352 R41 R31 R1
N=300
AB
N=200
AC
N=200
BC
N=0
AW
N=0
BW
N=0
CW
Figure 9. Stability regions of lamellae-in-lamellae for A(BC)3 linear multiblock copolymer thin films. The copolymers are the same as those in Figure 8, except for the solid substrates. The solid substrates are non-selective to the A-, B-, and C-blocks by setting xAWN ¼ xBWN ¼ xCWN ¼ 0.
N=300
L71
4.0
4.0
A B C
R472
AB
10.0
L1
10.0 6.0
18.0
L72
L52
12.0 8.0
L72
L52
AC
6.0
18.0
N=300
L32
substrates and seven BCBCBCB-layers in the middle (R372), and doubly periodic parallel lamellae-in-lamellae with four BCBC-layers near substrates and seven BCBCBCB-layers in the middle (R472). Note that we introduced another bold letter R to replace the bold letter L in the representations such as L51. In the representation such as R472, the first and second subscripts denote the number of small-length-scale lamellae near two substrates and in the middle, respectively. It can be seen that the phase transition of R?1 ! R31 ! R?1 ! R41 ! R352 ! R372! R472 takes place with increasing the film thickness. Similar to the thin films with the substrates selective to A-blocks, the number of largeand small-length-scale lamellae and the orientation of small-length-scale lamellae can be tuned by varying the film thickness. For the double-period lamellae-in-lamellae, the numbers of small-length-scale lamellae near the substrates and in the middle can be varied, respectively. From the diagram, we also noted the reentrant phase transition of R?1 and the disappearance of R?2, which is similar to the cases with substrates selective to A-blocks (see Figure 8). In the present work, we examined the stability of lamellae-in-lamellae in the thin films confined between two substrates. Except for the parallel-to-perpendicular lamellar transition and the variation of the number of largeand small-length-scale lamellae, the reentrant phase transitions of perpendicular lamellae-in-lamellae were found. The phenomenon of reentrant phase transitions was not revealed in our previous DPD simulations on the hierarchical lamellae formed in thin films confined between two substrates, probably due to narrower regions of perpendicular lamellae-in-lamellae at relatively larger film thickness.[17] In addition, the present work also revealed that various new reverse lamellae-in-lamellae such as R?1 and R31 can be formed in the thin films between two substrates nonselective to A-, B-, and C-blocks. Such structures were not studied in our previous work by the
Macromol. Theory Simul. 2015, 24, 468–476 © 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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DPD simulations. Overall, the present work can complement our previous work well from the viewpoint of structure stability.[17] At this point, we aim to emphasize that the ‘‘masking’’ technique developed by Fredrickson et al. is very suited for the study of hierarchical microstructures in thin films.[33] In the ‘‘masking’’ technique, the linear multiblock copolymers are confined between two walls by choosing a ‘‘masking’’ function to fit the geometry of the walls. The ‘‘masking’’ technique can retain the stability characteristics of standard saddle point search methods. Moreover, using this technique, the self-consistent field equations can be readily solved with periodic boundary conditions. In particular, the numerical implement of the diffusion equations by the fourth-order backward differentiation formula can be as effective as those for the bulk. Therefore, the computational cost is essentially as the same as that for the bulk. What is more important is that the work can be conveniently extended to more complicated geometrical confinement. We also want to emphasize that the thin films of linear multiblock copolymers show some behaviors that are different from the thin films of widely-studied copolymers such as diblock copolymers. In thin films of lamellaeforming diblock copolymers, the perpendicular and parallel lamellae appear alternatively as the films become thick. The orientation (parallel or perpendicular) of lamellae is largely controlled by the condition of commensurability between the film thickness and the bulk period of the lamellae. However, this principle could not apply to the present cases. For example, in the phase transition of L?1 ! L51 ! L?1 ! L71 ! L?2 ! L52! L?2 ! L72, the bulk phases (perpendicular lamellae-in-lamellae) are stable as the effective film thicknesses are in the range of 
